Kinetics of the intraband absorption and magnetoabsorption coefficients in mixed semiconductors with composition fluctuations

P. J. FLATAIJ: Kinetics of the Intraband Absorption and Magnetoabsorption 251 phys. stat. sol. (b) 90, 251 (1978) Subject classification: 1.3 and 20.1 ; 22.4.4 Institute of Theoretical Physics, Warsaw University1) Kinetics of the Intraband Absorption and Magnetoabsorption Coefpicients in Mixed Semiconductors with Composition Fluctuations BY P. J. PLATAU~) The influence of short-range ordering in a mixed semiconductor on the intraband absorption and magnetoabsorption coefficients is determined. The coefficients are shown to be temperature- dependent. Their time-evolution after changing the temperature is calculated. An experiment, permitting measurements of the degree of ordering (non-ideality) in alloys is proposed, and a numer- ical example for HgTe-CdTe is given. On determine l’effet d’ordre proche dens un semiconducteur mixte sur les coefficients d’absorp- tion intrabande et de magneto-absorption. On montre que ceux-ci dependent de la tempbrature, et on calcule leur Bvolution dam le temps provoqube per des variations thermiques. L’auteur pro- pose une experience ayant pour de mesurer le degre d‘ordre (#imperfection) des alliages. A titre d’exemple, on donne des calculs numeriques pour le systhme HgTe-CdTe. ’ 1. Introduction The interest in mixed semiconductor crystals has increased steadily over the past few years. They differ essentially from other semiconductors in that the distribution of atoms in their lattice is, to a higher or lesser degree, random, leading to local deviations in compositions v = v(r) from the average v,. These fluctuations can be expected to lead to various new, interesting effects unobserved in ideal structures. Alloy scattering [l], which occurs in mixed crystals in addition to the normal scat- tering mechanisms specific to non-mixed ones, is an effect of this kind. Recently, composition fluctuations have been shown to affect magnetoabsorption [2] and ab- sorption [3] on the free carriers.8) Studies on fluctuational effects in mixed semiconductors are commonly based on the assumption of a strictly random distribution of atoms in the lattice. However, numerous results for metal alloys point to the presence, usually, of some degree of ordering in mixed crystals. The deviations from strict randomness (the so-called non- ideality of the alloy) are related with differences in interaction and atomic radii of the components of the alloy. Eecent papers by Bublik et el. [a, 51 appear to provide evidence of short-range ordering in semiuonducting alloys as well. In the present paper, we shall be dealing with the influence of fluctuations in compo- sition on absorption and magnetoabsorption in mixed crystals, exhibiting a tendency to short-range order. Our considerations will also include non-equilibrium states, i.e. when the short-range ordering evolves in time. 2. Mixed Semiconducting Crystals We shall concentrate on triple substitutional II-b7I, 111-V, IV-T’I compounds of the A,Bl-,C type in which one of the elements (denoted as C) is shared by both I) 00-681 Warsaw, Poland. 2) Present address: Institute of Geophysics, Warsaw University, 02-093 Warsaw, Poland. In equation (39) of [Z] the factor 8 in the denominator should be replaced by 16. In equation (3.18) of [3] in line 4 the factor c8/12 should be replaced by C8/12.

252 P. J. YLATAV component,s of the crystal. In this situation, a C-atom can have, as all its nearest neighbours, atoms A or atoms B or a mixture of the ones and others. Thus the crystal can be divided into primitive cells of the type A (containing atoms C and A) and of the type B (with atoms C and B). As we shall have to deal with the statistical thermo- dynamics of mixed crystals, we proceed to give a definition of the lattice model: 1. We assume the potential energy of the crystal as the sum of t,he interaction energies of pairs AA, BB, and AB (configurational term) and a constant term, equal for all atomic configurations. 2. We shall be neglecting all pairwise (AA, BB, AB) interactions unless the part- ners are nearest neighbours (nearest neighbour model). This assumption is roughly correct for crystals in which long-range forces are unessential. 3. We assume the lattice parameters as independent of composition, and the thermal dilatation of the lattice as negligible. On the above assumptions, we write the configurational part U, of the energy in the form u, = NAAEAA + NABEBB + NBBEBB 5 where NAA, NBB, and NAB are, respectively, the numbers of nearest neighbours AA, BB, and AB, whereas EAA, EBB, EAB are the respective interaction energies. By assumption 3, we shall omit volume effects (differences in volume of primitive cells). However, these effects should, in some cases, be taken into account (Guggen- heini [S]). We now introduce the following quantity, providing a measure of the degree of non-ideality of the crystal: The change in configurational energy induced by the transposition A to (2) - Y- -- EAB - + (EAA + EBB) - B amounts AUC = ANABS. (3) The parameter c" has a simple physical meaning. If 9 = 0, an interchange of A and B does not affect the configurational energy of the crystal. We refer to this caae as that of an ideal solid solution. An ideal alloy exhibits no tendency to atomic ordering. If E" =t= 0, we deal with a non-ideal alloy which exhibits a tendency to short-range ordering. A binary alloy arises as the result of mixing of its two components. Often, it is desirable to compare the properties of the mixture to those of the pure components. To this aim, we define the following thermodynamical mixing functions: = x(p, y', 3, n2) - xA(p, T, - xB(P, T, 9 (4) where n, and n2 are the numbers of moles of A and B, respectively, whereas X is a thermodynamical potential. In order to separate mixing effects from those due to variations in temperature and pressure, we shall cornpale the mixture and its compo- nents at the same values of T and p. Dealing with non-ideal solutions, we shall conpider the simplest model of a devia- tion from ideality. We define the concept of a regular solution as follows. The entropy of mixing is the same as for an ideal crystal, and the energy of mixing is different from zero. An interchange of sites by the atoms of the alloy leads to a variation of internal energy. This description is justified provided c" is small. Using the assumptions 1 to 3, we obtain the following expression for the Gibbs free energy: where N is the number of lattice points and z is a coordination number. A,G = Nzw(1 - w) B + NkT [w In v + (1 - w) In (1 - w)] , (5) 15213951, 1978, 1, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/pssb.2220900127 by University of California San Diego, Wiley Online Library on [24/09/2026]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License

Kinetics of the Intraband Absorption and Magnetoabsorption Coefficients 253 Moreover, we calculate the derivative a2A,apv2: a2A,G N kT ~ = - 2zNE + v(l - v) av2 One notes that in the case of an ideal or weakly non-ideal solution the curve of A,G versus v is always concave (a2AmG/av2 > 0). Henceforth, we shall deal with weakly non-ideal solutions. Formula (5) was derived in the approximation of a regular solution and on the assumptions 1 to 3. Irrespective of these restrictions, formula (5) is found to be most useful in the description of a wide class of phenomena in solids, e.g. processes of nucle- ation [7], the kinetics of short-range ordering [8], critical scattering in alloys [9], and others. 3. Intraband Absorption and Magnetoabsorption in Mixed Semiconductors The present section is devoted to a discussion of the formulae for the magneto- absorption [2] and absorption [3] coefficients of mixed semiconductors. We restrict our analysis to a non-degenerate, parabolic and spherical conduction band edge and neglect spin effects. If Iv(r) - vol is small and V(T) varies slowly on the dimension of the unit cell, the position-dependent inverse effective mass and hand edge may be written in the form [lo] where mz-1 and Uo are the average values of the inverse effective mass and band edge, respectively, ao, Po, and yo are macroscopic constants of the material defined for compo- sition vo. On expanding the function V(T) in a Fourier series we obtain The sum denoted C' excludes vectors with q = 0. Since v(r) is real, we have e(--q) = e*W. ( 10) In the presence of the magnetic field we restrict our considerations to the quantum limit [Z], i.e. we assume that the electrons occupy states with n = 0 only and that k, < 4+, where kZp is given by with N, being the electron concentration and A the magnetic length: kz+ = n2A2N, (11) where H is the magnetic field. The quantum limit condition can be written in the form where woo is the cyclotron frequency eH 15213951, 1978, 1, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/pssb.2220900127 by University of California San Diego, Wiley Online Library on [24/09/2026]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License

264 P. J. FLATAU For incident light linearly polarized along the z-axis and taking into consideration only transitions permitted in the presence of concentration fluctuations, i.e. at o > wco, where o is the frequency of the incident light, the formula for the magneto- absorption coefficient takes the following form : and n’ 2 2 is the Landau level; we have, moreovei, used the following notations: (!e(q)12> = V le(q)I2 (19) with: V the volume of the crystal, le(q)I2 the average over all possible configurations, and x the dielectric constant. In the absence of magnetic field and when the electrons are contained within the Fermi sphere of the radius k, = (37~2N,)1/~, i.e. at strong degeneracy of the electron gas [3], the formula for absorption coefficient on free carriers takes the form where EF is the Fermi energy, 5 = hw/EF and EF < ho is assumed. transitions, the energy ho of the incident light has to be less than the energy gap, In both cases (absorption and magnetoabsorption), since we consider intraband hw< E,. (21) The formulae (15) and (20) were derived in the “virtual crystal approximation” using the effective-mass Hamiltonian which was derived in Leibler’s [ 101 paper. 15213951, 1978, 1, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/pssb.2220900127 by University of California San Diego, Wiley Online Library on [24/09/2026]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License

Kinetics of the Intraband Absorption and Magnetoabsorption Coefficients 255 By (15) and (20), in order to calculate the absorption coefficients, the dependence of ( le(q)12> on q has to be given in explicit form. In [2, 31 the calculations were per- formed assuming a strictly random distribution of atoms A and B in the sublattice. However, with regard to the differences between the compounds AC and BC (the components of the alloy), the distribution can be expected to be not strictly random. A procedure permitting in general the calculation of (1Q(q)I2) can be found in the thermodynamic theory of fluctuations. Hitherto, when referring to the mean value, we had in mind the one over various configurations of atoms at thermodynamic equilibrium. However, considerations on systems not at equilibrium and their mean values can lead to interesting results, as it will be shown in the next sections. 4. The Absorption Coefficients for Non-Ideal Semiconductors To calculate (l~(q)l2), we have recourse to the thermodynamical fluctuation theory [ll]. A macroscopic theory of this kind can be applied to volumes large as compared with the atomic volume. We thus have to restrict our considerations to fluctuations with large wavelengths. This restriction is consistent with the assumptions made when deriving formulae (16) and (20). In order to determine the fluctuation wave probability, we have to consider the variation in concentration in the entire volume of the crystal. In the macroscopic approximation, the concentration fluctuations from one point in the alloy to another are characterized by the continuous function It should be noted that, in our case, the fluctuations lead to a re-grouping of the atoms in various parts of the crystal, i.e. J h(r) dV = 0. By the thermodynamic theory, the probability density for the occurrence of a fluctuation in concentration is AF being the difference in total Helmholtz free energy in the presence and absence of the fluctuation, (24) with f the Helmholtz free energy per unit volume and fthe average off over the whole crystal. It can be shown ([11, 121) by expanding Sf = f - Fin a series in powers of w(r) and using the Fourier expansion (9) of the concentration, that AP = J (f - fi dV As shown by Cahn and Hillierd [13], for a regular solution K is of the order of c/ro, where 9 is given by (2) and ro characterizes the diihensions of the primitive cell. Since < 1, we obtain using (6) (in the present case, there is no need to distin- guish between AP and AG) that the second term of (25) can be omitted. Finally, using (25) and applying (6) and introducing t,he notations N, = NIP and L( T) = (1 - wo( 1 - wo) 2zE/kT)-l, we obtain - 15213951, 1978, 1, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/pssb.2220900127 by University of California San Diego, Wiley Online Library on [24/09/2026]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License

256 P. J. FLATAU We now proceed to a discussion of formula (26). At high temperatures one has An identical result has been obtained [2] on the assumption of a strictly random distribution. This could have been expected, since at high temperatures the atoms tend to be distributed randomly. Formula (27) is also obtained on making c" go over to zero, i.e. on assuming the alloy to be an ideal one. The mean value (37) is indepen- dent of temperature and q. Temperature does not affect the magnitude of the fluc- tuations. If B + 0, ( le(q)lz) depends on temperature significantly. At the limiting concentrations (vo = 0 and vo = 1) both (26) and (27) vanish, as could be expected. As stated, the absorption coefficients (15) and (20) were calculated in [2] and [3] on the assumption of a random distribution of atoms in the sublattice. Since (26) is independent of the wave vector q, it is easily shown that if short-range order in the crystal is taken into account within the framework of the regular solution approxima- tion, the magneto-absorption coefficient K,tll and the absorption coefficient r] calculat- ed in [2] and [3] are modified by the factor L( T) : 6. The Kinetics of Absorption Coefficients Hitherto, we dealt with situations of equilibrium. In solids, however, with regard to the relatively low mobility of the atoms, non-equilibrium situations can be ex- pected to occur more commonly than in other media. Such a situation can be created, e.g. by subjecting the crystal to a sudden change of temperature. Phenomenological theories of kinetics of ordering in mixed crystals show that in such a case <le(q, W> = [<le(a, 0)l2> - (le(q, ~)I~>I e-zE~h.t + (le(a, ~o)I~> , (30) where b is the interdiffusion coefficient [la]. (le(q, 0)l2) and (!@(a, o0)1~) are equilib- rium mean values at the initial temperature To and final temperature Tw, respec- tively. By (26), we finally obtain A description of this kind was introduced by Cahn [15] basing on Fick's equation and is at present extensively developed in a.pplication to metal alloys [S]. A strict treatment in terms of statistical physics based on the master equation is due to Langer [16]. With regard to metal alloys, various experiments aimed at checking the theory [17], including numerical ones [18], have been carried out. We start from formula (15) for the magnetoabsorption coefficient in the quantum limit. We put yo = 0, thus neglecting in (8) the term proportional to the Laphcian of 6v as compared with the one proportional to dv, and we use (31). Denoting 45 ;Is , p=-, Y y=- Ex = - and X =- 2mz n'hwco n'hwco h%;F hw 15213951, 1978, 1, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/pssb.2220900127 by University of California San Diego, Wiley Online Library on [24/09/2026]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License

Kinetics of the Intraband Absorption and Magnetoabsorption Coefficients 267 we obtain, after some algebra 1 t) + ~ (Ptn')2 Fo(X, t) nIe + I' 1 2 2 + ~ ( Ptn')2 E;@, t) +m + [El(Ptn'A) - E,(Ptn'B)] Po n' 1 n'(n' + 1) (X2 - 1) + + Ptn'[E2(Ptn'A) - E2(Ptn'B)] - n'2(X2 - 1)2 [ :6 where A = [Yl/z - (X + Y - 1)'~2]1~2 , B = [Y1l2 + (X + Y - 1)1/2]1/z , Pn(X, t) = Bn exp (- Ptn'B) - A" exp (- Ptn'd) (n = 0, 1, ...) . We have, moreover m and for n = 0 It can be shown that El 7 -Ei(-X) and E2(X) = Eo(X) - El(X). Obviously, at t = 0, K,,ll(t) is equal to its equilibrium value at the initial temper- ature To, whereas at t + 00, Kn$t) tends to its equilibrium value at the temperature T, : li,*ll(t = 0) = Kn*II(To) 9 (35) Kn*ll(t = 00) = Kn,II(Tm) (36) with &II(T) defined by (28). Formula (20), after changing the variables We proceed to consider the time evolution of the absorption coefficient q(t). 17 physic8 (b) 90/1 15213951, 1978, 1, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/pssb.2220900127 by University of California San Diego, Wiley Online Library on [24/09/2026]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License

258 P. J. FLATAU using (31) and integrating over y, takes the form where The analytical form of (37), though easy to obtain, is bulky and algebraically in- convenient. We now propose the following experiment. A mixed crystal at thermodynamic equilibrium at temperature To is suddenly cooled (or heated) to temperature T, and annealed at the latter during the time t. Xext, the crystal is placed in a low-temper- ature region (when diffusion is extremely slow) and t,he absorption coefficients are measured. This may be done for different times t. Then, we can obtain the time evolu- tion of absorption coefficients. 6. Numerical Example For a numerical example of magnetoabsorption we assume material parameters close to those for n-€€go.75Cdo.25Te [2] at low temperatures: v0 = 0.25, N, = 1.5 x x loz2 ~rn-~, x = 10, mi = 0.02m0, a, = -10, Do = 0.6 eV, N, = 6.8 x 1OI6 cm-3, H = 50 kG, EF = 14.8 meV, Awco = 29 meV, E, = 200 meV, and proceed to consider the case n' = 2. Moreover we haveil = 115 A, Y = 0.25. The energy range of incident radiation corresponds to X-values from the interval (0.75 to 1.3). We note that the quantum limit condition hwco > h2k$/2w$ is fulfilled, as well as the assumptions ho, The cell dimensions in HgTe-CdTe are almost equal (cr~d~~ = 6.48 A, aHgTa = = 6.46 A). Hence, in solid solutions of these compounds cortelations due to the volume effect can be thought to be negligible, in agreement with our lattice model assumption 3. According to results of Bublik et al. [4, 51 for crystals of Hgo.73Cdo.27!I'e and n'hwco < E,. Fig. 1. Time evolution of magnetoabsorption coefficient 15213951, 1978, 1, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/pssb.2220900127 by University of California San Diego, Wiley Online Library on [24/09/2026]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License

Kinetics of the Intraband Absorption and Magnetoabsorption Coefficients 269 5- h/EF--) S'hUJlE,--, Fig. 2. Absorption coefficient versus time, for B > 0. (Read q(t) instead of q(lp)) Fig. 3. Absorption coefficient versus time, for B < 0. (Read ~(t) instead of q(2l)) Hgo.66Cdo.arTe (i.e. for crystals similar to that considered by us), the alloy HgTe-CdTe exhibits a tendency to clustering. We take E = 0.0075 eV, which is of the order of Bublik's results. Also HgTe-CdTe has been shown to comply to the regular solution approximat ion. The data for the interdiffusion and selfdiffusion coefficients are fragmentary [19]. In our work, we assumed b = 10-ls cm3s-l. Finally, we take To = 660 "C and T, = = 300 "C. Kzl,(t) calculated with the preceding parameters is plotted in Fig. 1 for various annealing times t. For a numerical example of absorption we assume the same material parameters except for N,, which we now take as equal to N, = 2 x 10l6 cm+. The latter value presents the advantage that the composition fluctuation effects are comparable with effects due to ionized impurities. The range of the incident radiation energy corre- sponds to 5 from the interval (14, 27). Obviously, (1 - 5) < 0. Likewise, the condi- tion hw < E, is fulfilled. For the above values the time dependence of the absorption coefficient is plotted in Fig. 2. Were our crystal endowed with a tendency to ordering (S < 0), the evolution in time of its absorption coefficient would have the shape of the curves plotted in Fig. 3. 7. Conclusions If S > 0, the factor L(T) is larger than unity. Hence, the absorption coefficients for all values of time are larger than those of the ideal crystal. Magnetoabsorption and absorption increase from a minimum value at *he higher temperature To to a maximum value at the temperature T,. Similar time-dependent curves as those of Fig. 1 and 2 are obtained assuming E < 0. However, in this case the absorption coefficients decrease from a maximum value at To to a minimum value at T,. This behaviour of the absorption coefficients can provide the basis for experimental determinations of the type of ordering in the alloy. On inspection of (32) and Fig. 1 as well as (37) and Fig. 2 and 3 it is apparent that the distance between the curves for t = 0 and t = 00 is completely defined by the quantity L( To) - L( Tm), and, thus, by 5. Some information about the diffusion coefficient can also be obtained from the kinetics of the absorption and magnetoabsorption coefficients. Aelenozaledgements I would like to thank Prof. Dr. J. Mycielski for suggesting the problem and for his many helpful discussions. I would also like to thank A. Kurowska, M. Sc., for providing me with a copy of her Thesis, and K. Flatau, M. Sc., for the English trans- lation. 17. 15213951, 1978, 1, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/pssb.2220900127 by University of California San Diego, Wiley Online Library on [24/09/2026]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License

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